In above mean field theory, we only considered the \(C_3\)  rotation symmetry and the translation symmetry. Actually, a spin liquid should preserve all the point group symmetries of the spin Hamiltonian. However, in the mean field description of a QSL phase, the symmetry group of the mean-field Hamiltonian is the projective symmetry group (PSG) whose group elements are space group operations followed by \(SU(2)\)  gauge transformations[1]}. We will restrict to the PSG of the exactly solvable Kitaev point (called the Kitaev PSG) [2]}, [3]} in the following discussion.
